Hamiltonian systems

  A Hamiltonian system with d degrees of freedom satisfies Hamilton's equations of motion [2]:

$${\bf q}' = \nabla_{\bf p} H \quad,\quad {\bf p}' = - \nabla_{\bf q} H$$

${\bf q}$ and ${\bf p}$ represent the position and momentum respectively and are d-dimensional vectors. $\nabla_{\bf x}$ is the gradient operator taken w.r.t. ${\bf x}$, ${\bf x}'$ denotes the derivative of ${\bf x}$ with respect to time, t. $H=H\bigl({\bf q}(t),{\bf p}(t)\bigr)=H({\bf q},{\bf p})$ is the scalar-valued, autonomous (time-independent) Hamiltonian function. A Hamiltonian system is thus (necessarily) of even dimension (with m=2d in the autonomous form of (1)).

The Hamiltonian $H({\bf q},{\bf p})$ is time-invariant, i.e. it is a constant of the motion. When the Hamiltonian is interpreted as the energy of the system, time-invariance is equivalent to conservation of energy. To see this consider the chain rule: